A Spectral–Entropy Framework for the Mersenne Infinitude Problem: Multi-scale Density–0 Rigidity and Transduction Diagnostics

## Overview This preprint develops a **windowed operator–variational framework** for the Mersenne infinitude problem (Mersenne primes \ (Mₚ = 2ᵖ-1\) ). The paper is deliberately structured to **isolate exactly where genuinelyMersenne-specific arithmetic input is required**, while keeping the remaining pipeline rigorous and modular. The guiding idea is to treat “Mersenne-empty exponent windows” as an **exceptional set** controlled by anentropy–rigidity functional, once (i) a **mixing diagnostic** is available and (ii) a uniform **energy-gap**lower bound can be enforced through a multi-scale exceptional mask. --- ## What is new in v0. 9 - **Data recency tightened via an explicit GIMPS freeze-date. ** The catalogue check and the “true-positive” stress test are synchronized with the public GIMPS list (access date 2026-01-04; 52 known Mersenne-prime exponents; most recent exponent \ (p=136279841\), discovered 2024-10-12 and announced 2024-10-21). - **Graphical abstract included. ** A one-page diagram visualizes the multi-scale windowing, entropy dissipation, and the density–0 closure pipeline, emphasizing the location of the transduction barrier. - **Barrier/necessity statements polished. ** The paper explicitly records why band-limited factor diagnostics cannot, by themselves, certify Mersenne primality, and why a transduction-type input is internally required for unconditional closure within this framework. - **Exploratory empirical appendix strengthened. ** Pilot ROC/AUC and calibration (reliability) plots are provided on a bounded range (e. g. \ (p 10⁷\) ), together with robustness/negative-control language, to support the plausibility of the empirical transduction threshold \ (₄₌₏\) as a *screening* criterion. --- ## Core program (high-level) ### 1) Windowed exponent modelWe work on prime exponents \ (p\) in variable windowsT = [T, \;T+T^, (0, 1), \]with a finite probability space \ ( (T, mT) \) induced by the window. ### 2) Factor-admissible prime bands and incidence kernelWe introduce factor-admissible prime bands and construct a symmetric, band-limited incidence kernel\ (K^ (1) ₌, ₓ\) on \ (T\), designed to avoid degeneracy and to support conductance estimates afterMarkovization. ### 3) Sieve-to-mixing bridge (conditional) A conditional bridge is formulated: BV / large-sieve type control on incidence degrees and overlaps yieldsconductance bounds, and hence **polylogarithmic** spectral-gap / logarithmic-Sobolev (LSI) diagnostics for thelazy Markov chain \ (P₌, ₓ\). ### 4) Entropy–rigidity functional and density–0 closure (rigorous module) For an explicit exceptional mask \ (ET T\), the paper studies\G₌, ₓ[=H (|mT) +₁\| P₌, ₓ-\|₂²+₂ₓ²1₄ₓ\, dmT. \]Assuming (i) a mixing diagnostic and (ii) an energy-gap premise tied to a **multi-scale Mask A**, the paperproves a **logarithmic density–0 closure** result: the set of Mersenne-empty windows has logarithmic density \ (0\). ### 5) Transduction (open target) The only genuinely Mersenne-specific obstacle is isolated as a **transduction hypothesis**: a quantitative implication asserting that sufficiently strong diagnostics force the existence of a Mersenneprime in the window. v0. 9 emphasizes this step as the remaining “last mile” for unconditional infinitudewithin the present program. --- ## Numerical appendix (exploratory, not a proof substitute) Numerics are used strictly to validate diagnostics (gap estimators, band stability, robustness of rescaledquantities such as \ (T² T\), and pilot threshold calibration). They are **not** used toreplace the transduction or energy-gap steps. --- ## Scope and non-claims - This paper does **not** claim an unconditional proof of infinitely many Mersenne primes from classical sieve bounds alone. - The barrier results are **methodological**: they do not claim formal independence from ZFC/PA. - Mellin–trace identities, when mentioned, are used only as **bookkeeping** for windowed diagonal observables, not as an analytic-continuation route to infinitude. --- ## Suggested citation Lee, Byoungwoo. *A Windowed Operator–Variational Framework for the Mersenne Infinitude Problem: Multi-scale Density–0 Rigidity, Transduction Diagnostics, and Methodological Barriers* (v0. 9). Zenodo, 2026. --- ## Keywords Mersenne primes; analytic number theory; sieve methods; large sieve; Bombieri–Vinogradov;Markov chains; conductance; spectral gap; logarithmic Sobolev inequality; entropy dissipation;rigidity functional; multi-scale windows; exceptional sets; logarithmic density; ROC/AUC; calibration;GIMPS; reproducibility.